High Mach Number Regime Research Articles

Electron Heating in High Mach Number Collisionless Shocks.

The energy partition in high Mach number collisionless shock waves is central to a wide range of high-energy astrophysical environments. We present a new theoretical model for electron heating that accounts for the energy exchange between electrons and ions at the shock. The fundamental mechanism relies on the difference in inertia between electrons and ions, resulting in differential scattering of the particles off a decelerating magnetically dominated microturbulence across the shock transition. We show that the self-consistent interplay between the resulting ambipolar-type electric field and diffusive transport of electrons leads to efficient heating in the magnetic field produced by the Weibel instability in the high Mach number regime and is consistent with fully kinetic simulations.

On the Over-Prediction of Centrifugal Compressor Pressure Ratio in the High Impeller Tip Mach Number Regime

Abstract Reynolds-averaged Navier–Stokes (RANS) simulation is a routinely used tool for turbomachinery research and development, but it often over-predicts the pressure ratio of centrifugal compressors especially in the high impeller tip Mach number regime. In this work, the effects of a series of geometrical and numerical uncertainties and errors on the aerodynamic performance of a centrifugal compressor are investigated systematically. The investigated compressor is the NASA CC3 centrifugal compressor under different impeller tip Mach number conditions. The investigated geometrical/numerical factors include the impeller blade fillet, the impeller hub cavity, the impeller running tip clearance, the averaging method in the post-process, the turbulence model, the inlet duct hub rotation, and the inlet turbulence boundary condition. Results show that the uncertainty of the predicted total pressure ratio generally increases with the impeller tip Mach number. Among the investigated factors, the averaging method, the impeller blade fillet, the turbulence model, and the impeller hub cavity have the most pronounced effects in determining the compressor total pressure ratio. By adopting realistic geometric features, advanced turbulence modeling treatments, and the same averaging method as the experiment, the over-prediction in the total pressure ratio can be alleviated. Detailed flow mechanism analysis with respect to the impeller hub cavity and the turbulence model has been performed. These findings provide valuable guidance for future RANS simulations of centrifugal compressors.

Kinetic modeling of fractal aggregate mobility

Although the mobility or transport parameters, such as lift drag and pitching moments for regular-shaped particulates, are widely studied, the mobility of irregular fractal-like aggregates generated by the aggregation of monomers is not well understood. These particulates which are ubiquitous in nature, and industries have very different transport mechanisms as compared to their spherical counterpart. A high-fidelity direct simulation Monte Carlo (DSMC) study of two fractal aggregates of different shapes or dimensions is undertaken in the slip and transitional gas regime to understand the underlying mechanism of gas-particle momentum transfer that manifests as the orientation-averaged mobility parameters of the particulates. The study specifically focuses on the viscous contribution of these parameters and develops a non-linear correlation for drag and lift parameters p and q obtained from DSMC by normalizing the axial and lateral forces. The drag parameter p predicted a monotonic increase in fractal particulate drag with respect to a spherical monomer while the lift parameter q shows an initial increasing trend but a decreasing tendency toward the high Mach number or high compressibility regime. The approximate model that captures the compressibility and rarefaction effects of the fractal mobility is used to study the evolution of these particulates in a canonical Rankine vortex to illustrate the wide disparity in the trajectories of the fractal aggregate vs a spherical geometry approximation generally found in the literature.

High order semi-implicit weighted compact nonlinear scheme for the all-Mach isentropic Euler system

The computation of compressible flows at all Mach numbers is a very challenging problem. An efficient numerical method for solving this problem needs to have shock-capturing capability in the high Mach number regime, while it can deal with stiffness and accuracy in the low Mach number regime. This paper designs a high order semi-implicit weighted compact nonlinear scheme (WCNS) for the all-Mach isentropic Euler system of compressible gas dynamics. To avoid severe Courant-Friedrichs-Levy (CFL) restrictions for low Mach flows, the nonlinear fluxes in the Euler equations are split into stiff and non-stiff components. A third-order implicit-explicit (IMEX) method is used for the time discretization of the split components and a fifth-order WCNS is used for the spatial discretization of flux derivatives. The high order IMEX method is asymptotic preserving and asymptotically accurate in the zero Mach number limit. One- and two-dimensional numerical examples in both compressible and incompressible regimes are given to demonstrate the advantages of the designed IMEX WCNS.

The Effect of Chemical Reactivity on the Formation of Gaseous Oblique Detonation Waves

High-fidelity numerical simulations using a Graphics Processing Unit (GPU)-based solver are performed to investigate oblique detonations induced by a two-dimensional, semi-infinite wedge using an idealized model with the reactive Euler equations coupled with one-step Arrhenius or two-step induction-reaction kinetics. The novelty of this work lies in the analysis of chemical reaction sensitivity (characterized by the activation energy Ea and heat release rate constant kR) on the two types of oblique detonation formation, namely, the abrupt onset with a multi-wave point and a smooth transition with a curved shock. Scenarios with various inflow Mach number regimes M0 and wedge angles θ are considered. The conditions for these two formation types are described quantitatively by the obtained boundary curves in M0–Ea and M0–kR spaces. At a low M0, the critical Ea,cr and kR,cr for the transition are essentially independent of the wedge angle. At a high flow Mach number regime with M0 above approximately 9.0, the boundary curves for the three wedge angles deviate substantially from each other. The overdrive effect induced by the wedge becomes the dominant factor on the transition type. In the limit of large Ea, the flow in the vicinity of the initiation region exhibits more complex features. The effects of the features on the unstable oblique detonation surface are discussed.

H∞/LQR Optimal Control for a Supersonic Air-Breathing Missile of Asymmetric Configuration

Abstract Robust control is challenging to achieve for air-breathing missiles operating in a high Mach number regime, such as at high supersonic speeds (M > 3). The challenge arises because of strong couplings, significant non-linearities and large uncertainties in the aerodynamics and propulsion system. The feasibility of achieving robust control in such applications is strongly linked to the development of an appropriate control design structure. The purpose of this paper is to illustrate that in order to stabilise a highly unstable airframe and achieve the required performance, a hybrid of two control schemes may be used to achieve best results. A state feedback linear quadratic regulator is used to stabilise the plant and a forward path H∞ optimal controller is used to achieve the required performance and robustness. We also highlight the complementary attributes of the two control schemes that together can generate a more robust controller; LQR is used since it can achieve good gain and phase margins, whereas, the H∞ control method is better equipped to deal with uncertainties.

The Dynamics of Very High Alfvén Mach Number Shocks in Space Plasmas

Abstract Astrophysical shocks, such as planetary bow shocks or supernova remnant shocks, are often in the high or very-high Mach number regime, and the structure of such shocks is crucial for understanding particle acceleration and plasma heating, as well inherently interesting. Recent magnetic field observations at Saturn’s bow shock, for Alfvén Mach numbers greater than about 25, have provided evidence for periodic non-stationarity, although the details of the ion- and electron-scale processes remain unclear due to limited plasma data. High-resolution, multi-spacecraft data are available for the terrestrial bow shock, but here the very high Mach number regime is only attained on extremely rare occasions. Here we present magnetic field and particle data from three such quasi-perpendicular shock crossings observed by the four-spacecraft Cluster mission. Although both ion reflection and the shock profile are modulated at the upstream ion gyroperiod timescale, the dominant wave growth in the foot takes place at sub-proton length scales and is consistent with being driven by the ion Weibel instability. The observed large-scale behavior depends strongly on cross-scale coupling between ion and electron processes, with ion reflection never fully suppressed, and this suggests a model of the shock dynamics that is in conflict with previous models of non-stationarity. Thus, the observations offer insight into the conditions prevalent in many inaccessible astrophysical environments, and provide important constraints for acceleration processes at such shocks.

A pressure-based Mach-uniform method for viscous fluid flows

ABSTRACTA pressure-based, Mach-uniform method is developed by combining the SLAU2 numerical scheme and the higher temporal order pressure-based algorithm. This hybrid combination compensates the limitation of the SLAU2 numerical scheme in the low-Mach number regime and deficiencies of the pressure-based method in the high-Mach number regime. A momentum interpolation method is proposed to replace the Rhie-Chow interpolation for accurate shock-capturing and to alleviate the carbuncle phenomena. The momentum interpolation method is consistent in addition to preserving pressure–velocity coupling in the incompressible limit . The postulated pressure equation allows the algorithm to compute the subsonic flows without empirical scaling of numerical dissipation at low-Mach number computation. Several test cases involving a broad range of Mach number regimes are presented. The numerical results demonstrate that the present algorithm is remarkable for the calculation of viscous fluid flows at arbitrary Mach number including shock wave/laminar boundary layer interaction and aerodynamics heating problem.

Linearized Euler Equations for the Determination of Scattering Matrices for Orifice and Perforated Plate Configurations in the High Mach Number Regime

The interaction of a plane acoustic wave and a sheared flow is numerically investigated for simple orifice and perforated plate configurations in an isolated, non-resonant environment for Mach numbers up to choked conditions in the holes. Analytical derivations found in the literature are not valid in this regime due to restrictions to low Mach numbers and incompressible conditions. To allow for a systematic and detailed parameter study, a low-cost hybrid Computational Fluid Dynamic/Computational Aeroacoustic (CFD/CAA) methodology is used. For the CFD simulations, a standard k–ϵ Reynolds-Averaged Navier–Stokes (RANS) model is employed, while the CAA simulations are based on frequency space transformed linearized Euler equations (LEE), which are discretized in a stabilized Finite Element method. Simulation times in the order of seconds per frequency allow for a detailed parameter study. From the application of the Multi Microphone Method together with the two-source location procedure, acoustic scattering matrices are calculated and compared to experimental findings showing very good agreement. The scattering properties are presented in the form of scattering matrices for a frequency range of 500–1500 Hz.

A design of a two-dimensional, supersonic KH experiment on OMEGA-EP

An experiment meant to investigate the evolution of single mode Kelvin–Helmholtz (KH) instability in the supersonic regime is presented and theoretically analyzed. This experiment is intended to provide a direct measurement of the two-dimensional vortex evolution so that the high-Mach-number effects can be measured. The proposed design takes advantage of the ability of OMEGA-EP to drive experiments for up to 30 ns to produce steady conditions for KH that endure long enough to observe substantial growth. KH growth for the proposed design has been analyzed using two-dimensional numerical simulations. The results were compared to synthetic temporal KH numerical simulations using non-dimensional scaling in the low and high Mach number regime. The comparisons show that the growth in the high Mach number regime is expected to be suppressed by up to a factor of two. The effects of two-dimensional rarefactions from the lateral boundaries of the experimental system were also investigated. It was found that they introduce no major uncertainties or hazards to the experiment. We produced simulated radiographs, which show that the proposed experimental system will enable observation of the KH structures. An experiment of this kind has not yet been performed, and therefore would serve to validate numerical results and analytical models presented here and in the literature.

Electron heating at Saturn's bow shock

Collisionless shock waves are a widespread phenomenon in both solar system and astrophysical contexts. The nature of energy dissipation at such shocks is of particular interest, especially at high Mach numbers. We use data taken by the Cassini spacecraft to investigate electron heating at Saturn's bow shock, one of the strongest collisionless shocks encountered by spacecraft to date. Measurements of the upstream solar wind ion parameters are scarce due to spacecraft pointing constraints and the absence of an upstream monitor. To address this, we use solar wind speed predictions from the Michigan Solar Wind Model. Since these model predictions are based on near-Earth solar wind measurements, we restrict our analysis to bow shock crossings made by Cassini within ±75 days of apparent opposition of Earth and Saturn. An analysis of the resulting set of 94 crossings made in 2005 and 2007 reveals a positive correlation between the electron temperature increase across the shock and the kinetic energy of an incident proton, where electron heating accounts for between ∼3% and ∼7% of this incident ram energy. This percentage decreases with increasing Alfven Mach number, a trend that we confirm continues into the hitherto poorly explored high–Mach number regime, up to an Alfven Mach number of ∼150. This work reveals that further studies of the Saturnian bow shock will bridge the gap between the more modest Mach numbers encountered in near-Earth space and more exotic astrophysical regimes where shock processes play central roles.

Mach number dependence of electron heating in high Mach number quasiperpendicular shocks

The efficiency of electron heating through microinstabilities generated in the transition region of a quasiperpendicular shock for a wide range of Mach numbers is investigated by utilizing particle-in-cell (PIC) simulation and model analyses. In the model analyses saturation levels of effective electron temperature as a result of microinstabilities are estimated from an extended quasilinear (trapping) analysis for relatively low (high) Mach number shocks. Here, modified two-stream instability (MTSI) is assumed to become dominant in low Mach number regime, while Buneman instability (BI) is assumed to become dominant in high Mach number regime. It is revealed that Mach number dependence of the effective electron temperature in the MTSI dominant case is essentially different from that in the BI dominant case. The effective electron temperature through the MTSI does not depend much on the Mach number, although that through the BI increases with the Mach number as in the past studies. The results are confirmed to be consistent with the PIC simulations both in qualitative and quantitative levels. The model analyses predict that a critical Mach number, above which a steep rise in electron heating rate occurs, may arise at the Mach number of a few tens.

Effect of strong thermalization on shock dynamical behavior

The dynamics of the perpendicular shock front is examined under various plasma parameters by using particle‐in‐cell numerical simulation. As widely accepted, above the critical Mach number (∼3) the front of (quasi‐)perpendicular shocks show nonstationary behavior due to the shock self‐reformation. In much higher Mach number regime (MA > 20), we find that dynamics of the shock front self‐reformation can be modified. Nonlinear evolution of microinstabilities in the shock transition region results turbulent profiles in a microscopic view (≤c/ωpe), while, from a macroscopic view (>several c/ωpe) because of rapid, strong thermalization in the shock transition region, the localized accumulation of the plasma due to ion dynamics is smeared out in both of the velocity phase space and real space. As a result, the shock self‐reformation is realized within a reduced time and space. We can say there is a possibility that rapid, strong dissipation helps to stabilize the macroscopic shock front dynamics; the shock self‐reformation still persists, though. The strong thermalization is caused by the nonlinear evolution of two‐stream instability between the electron and the reflected/incident ion components and following ion‐acoustic instability. We think that the modification of the shock self‐reformation process observed in high Mach number regime indicates an important role of electron kinetics and heating in the macroscopic shock front behavior.

Nonthermal Electrons at High Mach Number Shocks: Electron Shock Surfing Acceleration

We study the suprathermal electron acceleration mechanism in a perpendicular magnetosonic shock wave in a high Mach number regime by using a particle-in-cell simulation. We find that shock surfing/surftron acceleration producing the suprathermal electrons occurs in the shock transition region where a series of large amplitude electrostatic solitary waves (ESWs) are excited by Buneman instability under the interaction between the reflected ions and the incoming electrons. It is shown that the electrons are likely to be trapped by ESWs, and during the trapping phase they can be effectively accelerated by the shock motional/convection electric field. We discuss that suprathermal electrons can be accelerated up to $m_i c^2 (v_0/c)$, where $m_i c^2$ is the ion rest mass energy and $v_0$ is the shock upstream flow velocity. Furthermore, some of these suprathermal electrons may be effectively trapped for infinitely long time when Alfv\'en Mach number $M_A$ exceeds several 10, and they are accelerated up to the shock potential energy determined by the global shock size.

Strong Electron Acceleration at High Mach Number Shock Waves: Simulation Study of Electron Dynamics

Electron-ion dynamics in a perpendicular magnetosonic shock wave in a high Mach number regime is studied by using the particle-in-cell simulation. It is shown that in the shock transition layer nonlinear evolution of two-stream instabilities plays an important role on the electron rapid heating and acceleration. As the shock Mach number greatly exceeds the critical Mach number, a series of large-amplitude, coherent electrostatic waves with the electron holes in phase space are excited by the two-stream instability between the reflected ions and the incident electrons in the shock transition layer. As the incident electrons are decelerated by the instability, other electrostatic waves grow in time by another two-stream instability between the incident ions and the decelerated incident electrons. The dynamic timescale of these instabilities is of the order of ω, where ωpe is the plasma frequency. The nonlinear interaction of these waves leads to the strong electron heating as well as the nonthermal high-energy electron acceleration in the shock transition layer.

Testing Astrophysical Radiation Hydrodynamics Codes with Hypervelocity Jet Experiments on the Nova Laser

Recent shock-tube experiments using the Nova laser facility have demonstrated that strong shocks and highly supersonic flows similar to those encountered in astrophysical jets can be studied in detail through carefully controlled experiments. We propose the use of high-power lasers such as Nova, Omega, Gekko, and the National Ignition Facility (NIF) to perform experiments on radiation hydrodynamic problems such as jets involving the multidimensional dynamics of strong shocks. High-power lasers are the only experimental facilities that can reach the very high Mach number regime. The experiments will serve as diagnostics of astrophysically interesting gasdynamic problems and could also form the basis of test problems for numerical algorithms for astrophysical radiation hydrodynamic codes. The potential for experimentally achieving a strongly radiative jet seems very good.

Waves and turbulence in high Mach number nearly parallel collisionless shocks

We have performed a number of one‐dimensional hybrid simulations (massless electrons, macroions) of almost parallel collisionless shocks (angle between the upstream magnetic field and shock normal of 5°) for medium and high Mach number shocks. One of the key questions we address is whether downstream convected upstream waves or waves locally generated at the interface of the incoming solar wind and the partially thermalized plasma are the main source of the downstream turbulence. In the medium, supercritical Mach number regime the small‐wavelength interface waves are damped in a transition region and contribute to downstream dissipation [Krauss‐Varban, 1995]. These waves are right‐hand polarized and propagate in the downstream rest frame toward the shock. In an Alfvén Mach number regime around MA ∼ 8 we find behind the shock ramp right‐hand polarized waves with both helicities. These are interpreted in terms of both the ion/ion resonant and the ion/ion right‐hand nonresonant instability occurring in the shock transition region. The latter occurs, according to linear theory, when the beam density is high and when the relative speed between the ion beam and the ambient ions is large. In this Mach number regime the shock‐produced waves have larger wavelengths and are less strongly damped. We suggest that they are subject to a parametric decay instability, resulting in an inverse cascade of short‐wavelength waves to longer ones farther downstream. In the very high Mach number regime, damping as well as decay of the shock‐produced waves is slow in comparison with the transit time from the shock to the magnetopause. The shock‐generated waves are then the dominant source of the downstream turbulence.